On Inverses and Linear Independence
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Ohio University / OhioLINK
2011
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ndltd-OhioLink-oai-etd.ohiolink.edu-ohiou13045200702021-08-03T05:47:00Z On Inverses and Linear Independence Moore, Jeremy S. Mathematics invertible basis invertible algebra fluid algebra fluidity <p>We study various connections between the notions of invertibility of elements and linear independence of subsets of algebras over (not necessarily commutative) rings. The main emphasis is on two such notions: invertible and fluid algebras.</p><p>We introduce a hierarchy of notions about algebras having a basis B consisting entirely of units. Such a basis is calledan invertible basis and algebras that have invertible bases are said to be invertible algebras. The conditions considered in that hierarchy include the requirement that for an invertible basis B, the set of inverses B<sup>-1</sup> be itself a basis, the notion that B be closed under inverses and the idea that B be closed under products under a slight commutativity requirement. Among other results, it is shown that this last property is unique of group rings. Many examples are considered and it is determined that the hierarchy is for the most part strict. For any field F not equal to F<sub>2</sub>, all semisimple F-algebras are invertible. Semisimple invertible F<sub>2</sub>-algebras are fully characterized. Likewise, the question of which single-variable polynomials over a field yield invertible quotient rings of the F-algebra F[x] is completely answered. Connections between invertible algebras and S-rings (rings generated by units) are also explored.</p><p>While group rings are the archetype of invertible algebras, this notion is general enough to include many other families of algebras. For example, field extensions and all crossed products (including in particular skew and twisted group rings) are invertible algebras. We consider invertible bases B such that for any two elements from B, a scalar multiple of their product belongs to B. Alternatively, one may consider invertible bases with the requirement that for every basis element, a scalar multiple of its inverse must also be in the basis. We refer to these algebras, respectively, as being scalarly closed under products and scalarly closed under inverses. We explore connections between these ideas and crossed products, twisted group rings, and skew group rings. We show that matrix rings over arbitrary rings are invertible algebras and also determine some types of infinite matrix rings which are invertible too.</p><p>We conclude the dissertation considering the property that sets of inverses of linearly independent invertible elements be also linearly independent. We refer to algebras with this property as fluid algebras. Fluidity of direct sums will be considered. We will characterize which single-variable polynomials over a field yield fluid quotient algebras of the F-algebra F[x]. We apply those results to establish when finite field extensions are fluid algebras. Also we will show that infinite field extensions are rarely fluid. We then define the fluidity of an R-algebra A to be an integer, such that for every set of n or less linearly independent invertible elements, their inverses are also linearly independent. The fluidity of various families of algebras such as matrix rings and field extensions is explored.</p> 2011-07-26 English text Ohio University / OhioLINK http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1304520070 http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1304520070 unrestricted This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws. |
| collection |
NDLTD |
| language |
English |
| sources |
NDLTD |
| topic |
Mathematics invertible basis invertible algebra fluid algebra fluidity |
| spellingShingle |
Mathematics invertible basis invertible algebra fluid algebra fluidity Moore, Jeremy S. On Inverses and Linear Independence |
| author |
Moore, Jeremy S. |
| author_facet |
Moore, Jeremy S. |
| author_sort |
Moore, Jeremy S. |
| title |
On Inverses and Linear Independence |
| title_short |
On Inverses and Linear Independence |
| title_full |
On Inverses and Linear Independence |
| title_fullStr |
On Inverses and Linear Independence |
| title_full_unstemmed |
On Inverses and Linear Independence |
| title_sort |
on inverses and linear independence |
| publisher |
Ohio University / OhioLINK |
| publishDate |
2011 |
| url |
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1304520070 |
| work_keys_str_mv |
AT moorejeremys oninversesandlinearindependence |
| _version_ |
1719425181782376448 |